Carrier frequency error estimation is often performed in a receiver of a communication system to eliminate offsets between a received signal's actual frequency and a frequency assumed by the receiver. An optimal maximum likelihood estimator (MLE) for the offset is given by the location of the peak of a spectral plot for the signal. Used with a discrete Fourier transform (DFT), the MLE has been shown to achieve the Cramer-Rao lower bound on variance at high signal-to-noise ratios (SNR).
A phase-locked loop (PLL) may be used for carrier phase tracking. Additionally, two cascaded PLLs may be used to form a carrier frequency error estimate. Phase estimates of a first PLL “converge” to a line whose slope is proportional to the offset between the actual and assumed carrier frequencies of the received signal, while phase estimates of the second PLL converge to a constant value corresponding to the phase offset.
Other carrier frequency error estimation methods involve the processing of de-spread data samples, e.g., algorithms devised for high capacity data radio (HCDR) narrowband mode. In HCDR, the complex conjugate of a known training sequence is multiplied by received de-spread data samples to generate an error signal. This removes the data modulation while leaving information about the frequency offset. The complex error signal is then rotated to produce samples of a complex exponential that, if plotted, would be centered about the real axis. This method then forms an estimate of the carrier frequency error by calculating the slope of the resulting line divided by an estimate of the signal amplitude.
In a relatively simpler approach, the frequency estimate of a complex sinusoid x(k) in complex white Gaussian noise is calculated according to equation 1:
                              Δω          =                                    ∑                              k                =                1                                            N                -                1                                      ⁢                                                  ⁢                                          w                k                            ⁢                                                          ⁢                              arg                ⁡                                  [                                      x                    *                                          (                      k                      )                                        ⁢                                          x                      ⁡                                              (                                                  k                          +                          1                                                )                                                                              ]                                                                    ,                            (        1        )            where N is the number of symbols and Wk is a weighting function given by equation 2:
                              w          k                =                                            6              ⁢                              k                ⁡                                  (                                      N                    -                    k                                    )                                                                    N              ⁡                              (                                                      N                    2                                    -                  1                                )                                              .                                    (        2        )            This well-known approach was developed by Steven M. Kay and is referred to as “Kay's approach.” In Kay's approach, the frequency is estimated using the weighted average of N−1 differential phase values, which are calculated as the arctangent of the product of the complex signal's conjugate and a time-shifted version of itself. This method has also been shown to achieve the Cramer-Rao lower bound for high SNR levels due to the properties of the weighting function.
Digital carrier frequency error estimation using the DFT approach described above may be too costly computationally for certain applications even if a fast Fourier transform (FFT) is used. Similarly, the PLL approach entails additional hardware and processing complexity that make this method undesirable compared to simpler algorithms. A limitation of the HCDR approach is the appearance of a bias error in the carrier frequency error estimate that increases with the degree of the offset and is independent of the symbol energy to noise energy (Es/No) level. The addition of a compensation factor may reduce the mean estimation error, however, this results in increased complexity along with an increase in the variance of the estimation error. With respect to Kay's approach, direct application to a Quasi-Bandwidth Limited Minimum Shift Keyed (QBL-MSK) modulated waveform illustrates that adequate performance requires a relatively large number of symbols. Accordingly, improved carrier frequency error estimation methods and apparatus are needed that are not subject to these limitations. The present invention addresses this need among others.